Question

If $f$ be continuous on $[a, b]$ and differentiable on the open interval $(a, b)$, then which of the following is/are correct?

• A. $f$ is strictly increasing in $[a, b]$, if $f^{\prime}(x)>0$ for each $x \in(a, b)$
• B. $f$ is strictly decreasing in $[a, b]$, if $f^{\prime}(x)<0$ for each $x \in(a, b)$
• C. $f$ is a constant function in $[a, b]$, if $f^{\prime}(x)=0$ for each $x \in(a, b)$
• D. All the above are correct

Answer 1

Answer: (D)

Mean value theorem, which states that, if $f:[a, b] \rightarrow R$ be a continuous function on $[a, b]$ and differentiable on $(a, b)$, then there exists somec in $(a, b)$ such that
$f^{\prime}(c)=\frac{f(b)-f(a)}{b-a}$
Let $x_1, x_2 \in[a, b]$ be such that $x_1 < x_2$, then there exists a point $c$ between $x_1$ and $x_2$ such that
$ f\left(x_2\right)-f\left(x_1\right) =f^{\prime}(c)\left(x_2-x_1\right) $
i.e., $ f\left(x_2\right)-f\left(x_1\right) >0 $
i.e., $ f\left(x_2\right) >f\left(x_1\right)$
Thus, we have
$x_1 < x_2 \Rightarrow f\left(x_1\right) < f\left(x_2\right) \text {, for all } x_1, x_2 \in[a, b]$
Hence, $f$ is strictly increasing function in $[a, b]$.
The proofs of part (b) and (c) are similar.
Hence, option (d) is correct.
Remarks
(i) $f$ is strictly increasing in $(a, b)$ if $f^{\prime}(x)>0$ for each $x \in(a, b)$.
(ii) $f$ is strictly decreasing in $(a, b)$ if $f^{\prime}(x) < 0$ for each $x \in(a, b)$.
(iii) If a function is strictly increasing or strictly decreasing in an interval $I$, then it is necessarily increasing or decreasing in I. However, converse need not be true.